Monte Carlo Chord Length Sampling for $d$-dimensional Markov binary mixtures

TL;DR

This paper extends the Monte Carlo Chord Length Sampling method to 2D and 3D Markov binary mixtures, analyzing its accuracy against reference solutions for particle transport in higher dimensions.

Contribution

It generalizes CLS to higher dimensions and evaluates its accuracy in complex geometries, addressing a gap in previous one-dimensional studies.

Findings

CLS shows good agreement with reference solutions in 2D and 3D for certain configurations.

Discrepancies depend on material properties and system dimensionality.

The work provides benchmarks for future validation of CLS in higher-dimensional media.

Abstract

The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore-Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using this algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Larsen and Pomraning, and extended by Brantley. In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the material properties of the benchmark specifications and of the system dimensionality.